Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach

Abstract

In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.